1. Field of the Invention
The present invention relates to a substructuring method for use, especially, in analysis of a large scale field, by which a field quantity corresponding to a structure is analyzed by dividing the structure into two or more substructures and calculating a solution respectively corresponding to the substructures separately.
2. Description of the Related Art
Hitherto, various kinds of devices, which use electron beams and ion beams, have been put to practical use. Each of these devices guides charged particles by using an electric field and a magnetic field and utilizes the charged particles for a predetermined purpose. An example of such an electron-beam device is illustrated in FIG. 9. In this figure, reference numerals 101 to 106 designate electrodes, to each of which a predetermined potential is provided. Thus a predetermined electric field (distribution) is formed in a space 200. Further, in the case of this example, a predetermined magnetic field is applied in an .alpha.-.beta. direction. Moreover, each of the electrodes 101 to 106 has a nearly rotationally symmetric shape with respect to an axis .alpha.-.beta.. Furthermore, the device is constructed in such a manner that electrons are emitted from an end portion 107 of the electrode 106. An emitted electrode moves in a trajectory designated by reference numeral 190 under the influence of forces caused by the electric and magnetic fields.
The control of a trajectory of a charged particle is important in designing such a device. Moreover, it is necessary for such a control operation to predict a trajectory of a charged particle by performing a numerical simulation. Roughly speaking, each numerical simulation consists of (1) the step of calculating field quantities corresponding to an electric field and a magnetic field and (2) the step of computing a motion of the charged particle.
The calculation of a field quantity is performed by using, for example, the finite element method, the boundary element method or the integral equation method. On the other hand, the computing of a motion of a charged particle is performed by using a method of calculating the distance traveled by the charged particle and the travelling speed of the charged particle in sequence every finite period .DELTA.t. There are various kinds of such methods. Further, a typical example of such methods is the Runge-Kutta method. Furthermore, such methods are named generically as a time integral method.
Next, a method of calculating an electric field (distribution) in a space 200 of FIG. 9 will be described hereinbelow. Further, each of the electrodes 101 to 106 is at a fixed potential. Moreover, on boundaries 121 to 125, the normal component of the electric field can be regarded as being approximately equal to 0. Such a boundary condition can be used without degrading the precision of calculating the electric field in the vicinity of the trajectory 190 of an electron.
In the case of using the finite element method in the calculation of the electric field, the interior of the space 200 is first divided into elements of small volume. Next, the predetermined values of the potentials are given to the electrodes 101 to 106 as known quantities. Further, zero is given to each of the other boundaries 121 to 125 as the value of the normal component of (the intensity of) the electric field. When performing finite element calculations in this way, the value of the potential at each of nodes of elements are obtained. According to the finite element method, the distribution of the potential in each of the elements is represented by a predetermined function of the potentials of the nodes of the corresponding element. The value of the intensity of the electric field, therefore, can be obtained by differentiating this intra-element function with respect to the position. Consequently, as long as a particle is within the space 200, the value of the intensity of the electric field at the position of this particle can be calculated.
However, high accuracy is sometimes needed for the calculation of the trajectory of a particle. In such cases, it is necessary to enhance both of the precision of calculating a field quantity corresponding to a field and that of computing the motion of a charged particle. Because of the fact that generally, the precision of calculating a field quantity is lower than that of computing the motion of a charged particle, the enhancement of the precision of calculating a field quantity becomes important. For that purpose, when performing the finite element method, the interior of the space has only to be divided further finely into elements. However, actually, there is a limit to the capability or throughput of a computer used for the calculation. Especially, storage or memory capacity, which is available for the calculation, imposes limitations on the magnitudes of field quantities to be calculated (or to the computational complexity). One of the methods for overcoming such limitations is the substructuring method.
The substructuring method is a method, by which a space to be analyzed is first divided into some substructures and next, one of the substructures is analyzed by applying the finite element method thereto, and finally, a solution to be found in the entire space is obtained by manipulating conditions established in each of the connecting portions among the substructures.
Next, the substructuring method will be specifically described hereinbelow by referring to FIG. 10. In this figure, reference numerals 101, 105, 106, 141 to 149 respectively denote electrodes; and 201 to 207 subspaces. Further, reference numerals 121 to 125 respectively designate boundaries between the entire space and the exterior, similarly as of the space of FIG. 9; and 131 to 136 boundaries between the subspaces into which the entire space is divided. The entire shape of the device is nearly rotationally symmetric with respect to an axis .alpha.-.beta., similarly as in the case of the device of FIG. 9. FIG. 10 is a sectional view of such a device.
Each of the substructures is formed as follows.
A first one of the substructures consists of the subspaces 201 and 202 and the boundaries 101, 106, 141, 142 and 132.
A second one of the substructures consists of the subspaces 202 to 204 and the boundaries 142 to 145, 123, 131 and 134.
A third one of the substructures consists of the subspaces 204 to 206 and the boundaries 145 to 148, 124, 133 and 136.
A fourth one of the substructures consists of the subspaces 206 and 207 and the boundaries 148, 149, 125, 135 and 105.
Therefore, the device of FIG. 10 is constituted by the four substructures. Each of the subspaces 202, 204 and 206 is an overlapping subspace between the adjoining substructures.
&lt;Example of Calculation of Electric Field&gt;
A process of calculating an electric field will be described hereinbelow by referring to a flowchart of FIG. 11.
The calculation is commenced in step S610. Next, in step S611, fixed boundary values, namely, the values of the potentials are set at nodes on the surfaces of the electrodes 101, 105, 106 and 141 to 149 as the fixed boundary values. Subsequently, in step S612, initial values are set at nodes on the boundaries 131 to 136 of the substructures. Here, the initial values are zero. However, in the case of the example of FIG. 10, the potential on the electrode in the vicinity of each of these boundaries may be set as the initial value to be set at a corresponding one of the nodes. Namely, the potentials on the electrodes 141 to 143 may be given to the nodes on the boundaries 131 and 132, respectively. Incidentally, as will be described later, practically, the initial values are not used. Thus, the setting of the initial values can be omitted. It is the same with the boundaries designated by reference numerals 133 and 135.
In the case of the calculation of a field (quantity) in one of the substructures to be step S613, first, such a calculation is performed on the first substructure (consisting of the subspaces) 201 and 202 according to the finite element method by using the boundaries 101, 106, 141, 142, 132, 121 and 122. Incidentally, in the cases of the boundaries 121 and 122, the boundary conditions are automatically met. Thus, it is unnecessary to set the initial values at the nodes on these boundaries 121 and 122.
In step S615, the value of the potential obtained as the result of the calculation performed on the first substructure is set at a node on the substructure boundary 131 contained in the first substructure. This means simply the replacement of the value at the node, because of the fact that the node in the overlapping subspace between the first and second substructures is common thereto.
A process consisting of steps S613 to S615 is performed on the first to fourth substructures in sequence. Next, such a process is again performed on the substructures by starting with the first substructure. In the aforementioned step S614, a judgment is made on midway convergence conditions in the repeated process. This judgment is performed so as to check the amount of the change in value of the potential at a node of a same substructure in a cycle of the repeated process, namely, the difference between the current value of the potential at the node and the preceding value thereof. Further, if the extent of the change in value of the potential at the node becomes sufficiently small, it is judged that the potential at the node has converged on the current value thereof. Moreover, if the convergence of the potentials at nodes is achieved in all of the substructures, it is judged that the potentials at the nodes has converged in the entire space to be analyzed. Then, the process or program exits from step S614 through a YES-branch to step S616 whereupon the calculation is finished.
&lt;Example of Calculation of Trajectory&gt;
Next, the procedure for calculating a trajectory of a charged particle will be described hereinbelow by referring to a flowchart of FIG. 12. Basically, this procedure is similar to that for calculating a trajectory of a charged particle in the case where the space to be analyzed has a single structure.
In step S620, the electric field is calculated. Namely, the electric field at each of all nodes is calculated according to the finite element method in this step. Then, the program advances to the next step.
In step S621, the value of the intensity of the electric field at each of the nodes is calculated. Further, the reason for calculating the value of the intensity of the electric field at each of the nodes is as follows. Here, it is assumed that the space to be analyzed is divided into elements as illustrated in FIG. 13 (incidentally, in this figure, the elements are shown as two-dimensional elements, for the simplicity of the description). The space of FIG. 13 is composed of elements 21 to 28 and nodes 11 to 19. Here, note that the electric field at an arbitrary point in the element 21 can be determined by the potentials at the nodes 11, 14 and 15 and an interpolation function defined in the element 21. The accuracy of calculating the electric field (distribution), however, becomes lower than that of calculating the potential field distribution. For example, in the case of a linear tetrahedron element, the value of the intensity of the electric field is constant therein but changes discontinuously on the boundaries thereof. This is extremely unnatural. Thus, if the value of the intensity of the electric field on each of the nodes is determined by some method and further, that of the intensity of the electric field in the element is obtained by interpolation from the values of the intensity of the electric field on each of the nodes. Thereby, the electric field can be further smoothly represented.
For instance, a method of defining the value of the intensity of the electric field at a node as the average of the values of the intensity of the electric field in an element containing a node is used as a method of calculating the value of the intensity of this node. If this method is applied to the space of FIG. 13, the value of the intensity of the electric field comes to be defined as the average of the values of the intensity of the electric field in the elements 21 to 23 and 26 to 28.
A loop S627 consisting of steps S622 to S624 is a portion or process for calculating the trajectory of a particle.
Further, reference character S622 designates a step in which the value of the intensity of the electric field at a position of a particle is calculated by interpolation by use of the values of the electric field at the nodes, which have been obtained in step S621, in the element. Then, in step S623, the equation of motion is solved. Moreover, the position and speed of the particle at a point of time when an infinitesimal time .DELTA.t has passed thence are computed. During a subprocess containing steps S622 and S623 and a path or branch P626 is repeatedly performed, the calculation corresponding to the particle is finished in step S624 when the particle reaches the final position thereof. Then, the program passes or moves on to the calculation of the trajectory of the next particle. When the calculation of the trajectories of all of the particles is finished, the procedure for calculating the trajectory is finished and the program advances to step S625.
&lt;Example of Space-Charge Effects&gt;
In the foregoing description, the calculation of the trajectories has been described. However, there is a problem to be discussed, which is left unmentioned, namely, a problem of a space-charge effect. Thus, the space-charge effect will be described hereinafter.
The space-charge effect is generally defined as a phenomenon in which the charge of a charged particle moving in a space causes an electric field. Namely, this means that when a charged particle is generated, the electric field (distribution) is changed from that produced only by the electrodes when there is no charge particle. This effect is large in a charged-particle emitting portion in which the speed of a charged particle is low, namely, in the proximity of the electron emitting portion 107 in the case of the device of FIG. 9. This effect becomes an important factor which not only causes the trajectory of the particle but also determines an emission current (incidentally, in the case of a field emission, the electric field occurring in the emitting portion changes owing to the presence of space charge and thus affects the emission current density directly). Therefore, it is necessary to calculate the trajectory of a charged particle by taking this effect into account.
A calculation procedure for performing the calculation in such a way will be described hereinbelow by referring to a flowchart of FIG. 14.
The calculation is started in step S630. Next, in step S631, the calculation of the electric field is performed. Here, note that if the influence or effect of a magnetic field caused by the motion of a charged particle can be neglected, the magnetic field does not affect the trajectory of a charged particle. Thus, it is sufficient to perform the calculation only one time. Then, in step S632, the trajectory of the charged particle is calculated.
Subsequently, if it is judged in step S633 that the value does not converge, the program advances to step S634 whereupon the space-charge distribution is calculated by using information concerning the position and speed of the charged particle. Next, the program returns through a path or branch P636 to step S631 whereupon the electric field is calculated again by using the space-charge density distribution which has been computed in step S634. Thereafter, the process consisting of steps S631 to S634 is repeated.
During repeating such a process, it is judged in midway step S633 whether or not the calculated values of the trajectory and so on have converged. If the convergence conditions are satisfied (namely, it is judged in step S633 that the calculated values have converged), the program exits therefrom to step S635 in which the calculation is then finished. Incidentally, in step S633, in the case where the trajectory of the charged particle, the value of the speed thereof and the value of the intensity of the electric field come to be able to be regarded as being unchanged, it is judged in step S633 that the calculated values have converged.
In the foregoing description, the conventional method of calculating the trajectory and so on of a charged particle placed in an electromagnetic field has been described. The conventional method, however, has encountered the following problems.
(1) In the case where high precision is required for the calculation of the trajectory of a charged particle, high accuracy is also required for the calculation of an electric field or a magnetic field. However, when performing the finite element method, an electric field (distribution) in an element is represented by a polynomial. Therefore, a large-scale analysis using the substructuring method is insufficient for accurately calculating the electric field (distribution).
(2) As the countermeasure to this, an analysis method, by which the electric field (distribution) can be further precisely represented or expressed, is considered to be employed. Examples of such an analysis method are the boundary element method and the integral equation method. Incidentally, these methods have problems in that the scale of an analysis becomes large and that the computational complexity, as well as the storage capacity used for the calculation, increases.
(3) Further, in the case where only one of such calculation techniques is employed, there is a drawback in that the accuracy of calculating the electric field in the neighborhood of a boundary, especially, in the vicinity of the electron emitting surface as designated by reference numeral 107 in FIG. 9 is low. Namely, the electric field occurring on the emission surface determines the emission current density and thus may cause a large error in the trajectory of a charged particle and in the amount of the emission current.
(4) Moreover, in the case where the space-charge effect is taken into consideration as described above, the space charge should be used in the calculation. In contrast with the finite element method by which a space is divided into elements and thus the electric charge distribution can be easily handled, the boundary element method and the integral equation method require special means or ideas for handling the electric charge distribution. Moreover, the boundary element method and the integral equation method have a defect in that if the handling of the electric charge distribution is inappropriate, the accuracy of calculating the electric field is degraded and thus such techniques lose their intrinsic advantage that the accuracy of calculating a field quantity is good.
The present invention is accomplished to eliminate the drawbacks of the conventional method.